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If it's an older rig then it's probably a component failure: MOBO, CPU, RAM, etc. The only way to 'start' is to breadboard and the easiest method it to remove the MOBO from the case and pull everything off leaving the bare minimum: MOBO, CPU, 1 Stick RAM, PSU and adding a GPU only if it stays on or doesn't have an onboard GPU.
Meaning don't connect the USB headers, Keyboard, mouse and short the PWR+ & PWR- {Ground} to start.
Failure:
IF MOBO, CPU, 1 Stick RAM, PSU fails try the other stick of RAM.
Failure:
Take to a shop, chances are it's either the MOBO or CPU so unless you have a spare CPU it's a best guess it's the MOBO.
A bad PSU can easily damage EVERYTHING it's connected to including MOBO, GPU, and to a lesser degree CPU and RAM.